# Section1.4Exercises

##### 1

Yes/No. For each of the following, write Y if the object described is a well-defined set; otherwise, write N. You do NOT need to provide explanations or show work for this problem.

1. $\{z \in \C \,:\, |z|=1\}$

2. $\{\epsilon \in \R^+\,:\, \epsilon \mbox{ is sufficiently small} \}$

3. $\{q\in \Q \,:\, q \mbox{ can be written with denominator } 4\}$

4. $\{n \in \Z\,:\, n^2 \lt 0\}$

##### 2

List the elements in the following sets, writing your answers as sets.

Example: $\{z\in \C\,:\,z^4=1\}$ Solution: $\{\pm 1, \pm i\}$

1. $\{z\in \R\,:\, z^2=5\}$

2. $\{m \in \Z\,:\, mn=50 \mbox{ for some } n\in \Z\}$

3. $\{a,b,c\}\times \{1,d\}$

4. $P(\{a,b,c\})$

##### 3

Let $S$ be a set with cardinality $n\in \N\text{.}$ Use the cardinalities of $P(\{a,b\})$ and $P(\{a,b,c\})$ to make a conjecture about the cardinality of $P(S)\text{.}$ You do not need to prove that your conjecture is correct (but you should try to ensure it is correct).

##### 4

Let $f: \Z^2 \to \R$ be defined by $f(a,b)=ab\text{.}$ (Note: technically, we should write $f((a,b))\text{,}$ not $f(a,b)\text{,}$ since $f$ is being applied to an ordered pair, but this is one of those cases in which mathematicians abuse notation in the interest of concision.)

1. What are $f$'s domain, codomain, and range?

2. Prove or disprove each of the following statements. (Your proofs do not need to be long to be correct!)

1. $f$ is onto;

2. $f$ is 1-1;

3. $f$ is a bijection. (You may refer to parts (i) and (ii) for this part.)

3. Find the images of the element $(6,-2)$ and of the set $\Z^- \times \Z^-$ under $f\text{.}$ (Remember that the image of an element is an element, and the image of a set is a set.)

4. Find the preimage of $\{2,3\}$ under $f\text{.}$ (Remember that the preimage of a set is a set.)

##### 5

Let $S\text{,}$ $T\text{,}$ and $U$ be sets, and let $f: S\to T$ and $g: T\to U$ be onto. Prove that $g \circ f$ is onto.

##### 6

Let $A$ and $B$ be sets with $|A|=m\lt \infty$ and $|B|=n\lt \infty\text{.}$ Prove that $|A\times B|=mn\text{.}$